 # Thermal Energy Formula

The thermal energy is basically the energy present in a system. It is responsible for the temperature of the system. The flow of thermal energy is heat. Thermodynamics, a whole branch of physics, specifically deals with how heat transfers between various types of systems. It also deals with how the work is done in the process. Learn thermal energy formula here.

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## Nature of Thermal Energy

Almost all the energy transfer that takes place in the real-world physical systems does so with efficiency less than 100% and the results in some thermal energy. This energy is generally in the form of low-level thermal energy. Here, low-level refers that the temperature having an association with the thermal energy is so close to that of the environment. It is possible to extract the work only when there is a difference in the temperature. However, the low-level thermal energy represents “the end of the road” of the transfer of the energy.

### Derivation

Specific Heat Capacity = $$\frac{thermal energy input}{(mass)\times (temperature change)}$$

To write this equation in symbols, we will use C for specific heat capacity, T for Temperature, and Et for thermal energy. But the equation involves not T itself but the change in T during the energy-input process. The standard symbol we use for the “change” is the Greek letter we call delta (∆), so the change in T is written ∆T. Similarly, thermal energy input is that amount by which the thermal energy changes, ∆Et. Using these abbreviations, our equation becomes:

$$C = \frac{\Delta E_{t}}{m.\Delta T}$$
Often it is useful for rearranging this equation to solve for the change in thermal energy:

$$\Delta E_{t} = m.C.\Delta T$$
For example, to increase the temperature of a 10-kg wooden chair from 20◦C to 25◦C, it would need an energy input of:

$$\Delta E_{t} = m.C.\Delta T = (10kg)(1700J/kg.^{\circ}C)(5^{\circ}C)=85,000 J.$$
(Notably, the official units of ‘C’ are joules/kg per degree Celsius.)

Although there’s nothing fundamentally special about water, it is essential to us humans. Moreover, our society uses quite a bit of energy for heating the water, for various purposes. It’s, therefore, a good idea to remember the value of specific heat capacity of water:

$$C_{water} = 4200\frac{J}{kg.^{\circ}C} = 1 \frac{kcal}{kg.^{\circ}C} = 1\frac{Btu}{lb.^{\circ}F}$$

Where the last two values follow from the definition of the kilocalorie and the definition of the Btu. Because the quantity of water is often measured by volume whereas not by mass, it’s also useful to know that a kilogram of water occupies a volume of one liter (a little over a quart), whereas a pound of water can occupy a volume of one pint (two cups, or half a quart). A gallon of water equals eight pints (pounds), or 3.8 liters (kilograms).

## Solved Examples on Thermal Energy Formula

#### Question

Suppose the person shown in Figure 1 pushes the box, maintaining a constant velocity. The box has a mass of 100 kg and moves through a distance of 100 m. The coefficient of kinetic friction between the box and floor is $$\mu\kappa$$ = 0.3. what amount of thermal energy is transferred to the box-floor system?

#### Solution:

Since the box is not accelerating (it has constant velocity) and the force on the box is in the same direction as the direction of motion (no vertical component), the net force due to the person is exactly balanced by the force due to friction. This force applied over the given distance gives the change in thermal energy of the system.
$$\Delta E_{T} = 0.3.9.81 m/s^{2}.100kg.100m$$
= 29.43 kJ

#### Question

Suppose the paddlewheel depicted in Figure 2 is rotated by an electric motor which is rated at 10 W output power for 30 minutes. What is the amount of thermal energy that is transferred to the water?

#### Solution:

In this system, all the energy eventually is transferred to the thermal energy of the water (assuming the heat capacity of the paddles is negligible).
$$E_{T}$$ = Power . Duration
=10W. (30. 60s)
= 18 kJ.

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